Ordinary Differential Equations/Bernoulli

An equation of the form

$${dy \over dx}+ f(x)y = g(x)y^n$$

can be made linear by the substitution $$z=y^{1-n}$$

Its derivative is

$${dz \over dx}=(1-n)y^{-n}{dy \over dx}$$

So that multiplying it by $$y^{-n}$$

The equation can be turned into

$${dy \over dx}y^{-n} + f(x)y^{1-n} = g(x)$$

Or

$${dz \over dx} + (1-n)f(x)z = (1-n)g(x)$$

Which is linear.

Jacobi Equation
The Jacobi equation

$$(a_1+b_1x+c_1y)(xdy-ydx)-(a_2+b_2x+c_2y)dy+(a_3+b_3x+c_3y)dx=0$$

can be turned into the Bernoulli equation with the appropriate substitutions.